3.1 General structure
3.1.2 Scaling and regulation of the sample size
As discussed in Chapter 2, Monte Carlo models consider a sample from a real particle population.
Therefore, the real particle system is scaled down using a scaling factor𝑆. Scaling reduces the number of particles considered in the simulation, but the properties such as size distribution are identical.
Chapter 3 Micro-scale modeling using a Monte Carlo method
update particle properties
execute event choose event calculate timestep
scale system
end finished?
no yes
start
micro-processesrun
Figure 3.1:General flow chart of the Monte Carlo algorithm used in this work.
3.1 General structure
Table 3.1:Overview of constant and variable parameters for layering and agglomeration processes in reality and the simulation.
Layering Agglomeration Parameter
real MC real MC
𝑁pp − − const. 𝑓(𝑡)
𝑁p const. const. 𝑓(𝑡) const.
𝑆 const. 𝑓(𝑡)
Using𝑆, any parameter changing with the size of the particle system can be scaled from the real system to the sample system and vice versa. In previous works focusing on agglomeration, see Terrazas-Velarde [15], Hussain [97], and Dernedde [132], the scaling factor is defined based on the number of primary particles in the sample and in the real system, respectively. In case of layering processes, the distinction between particles and primary particles is, though, unnecessary and therefore the scaling factor is calculated from the number of particles. The scaling factor for both processes is thus defined as:
𝑆 =
𝑁p,MC
𝑁p,real layering, 𝑁pp,MC
𝑁pp,real agglomeration.
(3.1)
As the presented simulation method is a constant number Monte Carlo algorithm, the size of the sample system must be regulated if the particle number changes during the simulation to ensure statistical accuracy and reasonable computation times. In a real layering process, the number of particles is constant if overspray and breakage of particles are neglected. As a result, once the real system is scaled down, the sample size and the scaling factor stay constant over time and no regulation is needed. However, in a real agglomeration process the number of primary particles stays constant, while the number of particles changes due to agglomeration and breakage of bridges. In this case, the term “particle” comprises single primary particles and agglomerates. In the simulation the number of particles is held constant by randomly deleting a particle in case of a breakage event, which would otherwise increase the number of particles by one. In case of an agglomeration event, which decreases the number of particles by unity, one randomly selected particle is copied. This leads to a constant number of particles and a variable number of primary particles in the simulation.
As a result, the scaling factor changes over time and must be re-calculated in each time step. Table 3.1 gives an overview of constant and variable parameters in agglomeration and layering processes.
In case of agglomeration, the number of primary particles𝑁pp,MCin the simulation used in Equa-tion (3.1) is equal to the number of particles at the beginning of the simulaEqua-tion since no agglomerates exist at this stage. As the simulation proceeds,𝑁pp,MC changes and is accessible by counting the number of primary particles of all agglomerates in the sample system. The number of primary
Chapter 3 Micro-scale modeling using a Monte Carlo method
102 103 104
0 1 2 3 4 5 6
𝑁p,MC [−]
Relativevalue[−]
standard deviation computation time
Figure 3.2:Influence of the number of particles considered in the simulation on the relative standard deviation and computation time. The values are normalized to the simulation results, in which a number of 2000 particles were used. The data is taken from Terrazas-Velarde et al. [147].
particles𝑁pp,realin the real system is constant and can be calculated using the bed mass, particle density, and the initial particle size distribution, assuming spherical particles:
𝑁pp,real =
∫∞ 0
𝑛d𝑥 with 𝑛 = 6𝑀bed 𝜋𝜚p
𝑞0
∫∞
0 𝑥3𝑞0d𝑥
. (3.2)
In case of layering, the number of particles𝑁p,MCin the simulation in Equation (3.1) stays constant.
The number of particles in the real system stays constant as well and can also be calculated using Equation (3.2).
The remaining parameter, which must be set, is the size of the sample system. The size of the sample can be adjusted by the number of particles considered in the simulation. In order to guarantee statis-tical accuracy, this value should not be too small and, at the same time, it should not exceed a certain value to prevent long computation times. An investigation regarding the influence of the number of particles in the simulation on both the results and the computation times has been performed by Terrazas-Velarde et al. [147]. Several simulations with different sample sizes were performed.
The computation time and standard deviation characterizing the variation of the simulation results compared to the case with the highest accuracy (i.e., the largest number of particles) were evaluated.
The results, which are normalized to the values for the case of 2000 particles, are shown in Figure 3.2.
Based on these results, Terrazas-Velarde et al. [147] suggest that the number of particles in the simu-lation should lie between 1000 and 2000 particles to ensure accuracy and reasonable computation times. Zhao et al. [130] also recommend a sample size in the order of 1000 particles. As a result, the number of particles considered in the simulation is set to 1000 in this thesis.
3.1 General structure
Mass conservation
For agglomeration the combination of sample size regulation and the definition of the scaling factor in Equation (3.1) only ensures mass conservation if the primary particles are equally sized. However, if the primary particle size is distributed, mass conservation is not fulfilled and the scaling factor needs to be re-defined, as shown in the following example.
In this example, an amount of 1000 single particles (no agglomerates) is considered and no agglom-eration or breakage takes place. Particles are only randomly copied and deleted, ensuring a constant number of particles. This is done for equally sized primary particles as well as distributed primary particle sizes. The parameters used in this example are given in Table 3.2. In case of distributed primary particle sizes, a normal distribution of particle diameter is used, which is created as shown in Appendix C. In case of equally sized particles, every time a particle replaces another the total mass of the sample stays constant. However, in case of distributed primary particle sizes, a smaller particle may replace a larger one or vice versa. As a result, the total mass of the sample changes, while particle number is still constant. This effect increases with increasing variation in particle size.
In Figure 3.3, the evolution of the relative sample mass (total mass of the primary particles in the sample divided by its initial value) for an increasing number of sample size regulations (copying and deleting) is shown. If the particles are equally sized, mass is conserved, while mass is lost in case of the distributed primary particle sample. In this example, approximately 2 % of the initial mass is lost after 1000 sample size regulations. Therefore, information about the number of primary particles in the sample system is not sufficient to ensure mass conservation since the size of the primary particles plays a role.
As Equation (3.1) only relies on the number of the primary particles, this effect will lead to errors in the simulation. A solution is to re-define the scaling factor and directly use the total mass of the primary particles:
𝑆 = 𝑀pp,tot,MC
𝑀pp,tot,real. (3.3)
In this way, mass conservation is fulfilled even if the primary particle size is distributed. The total mass of the primary particles in the real system corresponds to the bed mass and the total mass of the primary particles in the simulation is calculated using the size and density of each primary
Table 3.2:Simulation parameters used in the example to investigate mass conservation of the sample system.
Parameter Value Unit
Mean particle diameter𝑑10 0.6 mm Standard deviation𝜎x 0 and 0.06 mm Particle density𝜚p 2500 kg m−3 Number of particles𝑁p,MC 1000 −
Chapter 3 Micro-scale modeling using a Monte Carlo method
0 200 400 600 800 1000
0.97 0.98 0.99 1 1.01
Number of sample size regulations[−]
𝑀pp,tot,MC/𝑀pp,tot,MC,0[−]
𝜎x=0 mm 𝜎x=0.06 mm
Figure 3.3:Evolution of therelativesample massdepending onthe numberof sample size regulations for equally sized particles (𝜎x =0 mm) and distributed particle sizes (𝜎x=0.06 mm).
particle. Since in case of layering the regulation of the sample size is not necessary, Equation (3.1) may still be used even if the initial particle size is distributed. In this work, Equation (3.1) is used in the Monte Carlo model for layering and Equation (3.3) is used for the agglomeration model.